# Closed subsets #

This file defines the metric and emetric space structure on the types of closed subsets and nonempty compact subsets of a metric or emetric space.

The Hausdorff distance induces an emetric space structure on the type of closed subsets of an emetric space, called Closeds. Its completeness, resp. compactness, resp. second-countability, follow from the corresponding properties of the original space.

In a metric space, the type of nonempty compact subsets (called NonemptyCompacts) also inherits a metric space structure from the Hausdorff distance, as the Hausdorff edistance is always finite in this context.

instance EMetric.Closeds.emetricSpace {α : Type u} [] :

In emetric spaces, the Hausdorff edistance defines an emetric space structure on the type of closed subsets

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• EMetric.Closeds.emetricSpace =
theorem EMetric.continuous_infEdist_hausdorffEdist {α : Type u} [] :
Continuous fun (p : ) => EMetric.infEdist p.1 p.2

The edistance to a closed set depends continuously on the point and the set

theorem EMetric.isClosed_subsets_of_isClosed {α : Type u} [] {s : Set α} (hs : ) :
IsClosed {t : | t s}

Subsets of a given closed subset form a closed set

theorem EMetric.Closeds.edist_eq {α : Type u} [] {s : } {t : } :
edist s t =

By definition, the edistance on Closeds α is given by the Hausdorff edistance

instance EMetric.Closeds.completeSpace {α : Type u} [] [] :

In a complete space, the type of closed subsets is complete for the Hausdorff edistance.

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instance EMetric.Closeds.compactSpace {α : Type u} [] [] :

In a compact space, the type of closed subsets is compact.

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• =

In an emetric space, the type of non-empty compact subsets is an emetric space, where the edistance is the Hausdorff edistance

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• EMetric.NonemptyCompacts.emetricSpace =
theorem EMetric.NonemptyCompacts.ToCloseds.uniformEmbedding {α : Type u} [] :
UniformEmbedding TopologicalSpace.NonemptyCompacts.toCloseds

NonemptyCompacts.toCloseds is a uniform embedding (as it is an isometry)

theorem EMetric.NonemptyCompacts.isClosed_in_closeds {α : Type u} [] [] :
IsClosed (Set.range TopologicalSpace.NonemptyCompacts.toCloseds)

The range of NonemptyCompacts.toCloseds is closed in a complete space

In a complete space, the type of nonempty compact subsets is complete. This follows from the same statement for closed subsets

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In a compact space, the type of nonempty compact subsets is compact. This follows from the same statement for closed subsets

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In a second countable space, the type of nonempty compact subsets is second countable

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NonemptyCompacts α inherits a metric space structure, as the Hausdorff edistance between two such sets is finite.

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• Metric.NonemptyCompacts.metricSpace =
theorem Metric.NonemptyCompacts.dist_eq {α : Type u} [] :
dist x y =

The distance on NonemptyCompacts α is the Hausdorff distance, by construction

theorem Metric.lipschitz_infDist_set {α : Type u} [] (x : α) :
LipschitzWith 1 fun (s : ) =>
theorem Metric.lipschitz_infDist {α : Type u} [] :
LipschitzWith 2 fun (p : ) => Metric.infDist p.1 p.2